To ensure games run well on Linux either via Native Linux builds or Windows games with Proton, part of the magic is in the Steam Linux Runtime. A new version of it, the Steam Linux Runtime 4.0 was recently put up with some pretty big changes.
What’s the point of it? It ensures Steam and games run through Steam on Linux work properly across all the many different Linux distributions. Another secret Valve sauce for Linux. Well, not secret at all but you get my meaning I’m sure.
The runtime is for launching games, not Steam itself. You can check the runtime selection in Compatibility tab of Steam and of each game. If your Steam Flatpak install doesn’t work, the issue is likely somewhere else.
I’d suggest trying to launch the flatpak from the terminal and seeing if there’s any strange logging.